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1822  00204  5938 

UMAK  ivHAYYAM  AS 


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A  MATHEMATICIAN 

WILLIAM  EDWARD  STORY 


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OMAR  KHAYYAM  AS  A  MATHEMATICIAN 


THE  UNIVERSITY  LIBRARY 

UNIVERSITY  OF  CALIFORNIA,  SAN  DIEGO 

LA  JOLLA,  CALIFORNIA 


WILLIAM  EDWARD  STORY 


OMAR  KHAYYAM 

AS  A 

MATHEMATICIAN 


By  WILLIAM  EDWARD  STORY 

Professor  of  Mathematics 
CLARK   UNIVERSITY 
Worcester.  -  Majwchuietts 


Read  at  the  Annual  Meeting  of  the 

OMAR  KHAYYAM  CLUB  OF  AMERICA 

1918 


April  5.  1919 

Thit  if  one  of  an  edition  privately  printed  by  the  Rosemary  Pm* 
for  the  member*  of  the  Omar  Khayyam  Club  of  America. 

Limited  to  200  copies,  on  American  deckel-edged  linen  paper, 
bound  with  vellum  back  and  antique  paper  ode*. 

Copies  numbered  1  to  100  reserved  to  Professor  Story. 

100  copies  numbered  1R  to  lOORreterved  for  the  Rosemary  Frew. 

This  i*  No. 


60R 


OMAR  KHAYYAM 

AS  A  MATHEMATICIAN. 

IT  seems  to  be  commonly  assumed 
that  Omar  was  by  profession  an 
astronomer  and  that  with  him  pure 
mathematics  was  only  a  side  issue.  But 
it  should  be  observed  that  all  the 
earlier  philosophers  were,  as  the  name 
"philosopher"  implies,  lovers  of  learn- 
ing of  all  kinds;  such  a  lover  of  learn- 
ing Omar,  indeed,  seems  to  have  been. 
The  true  philosopher  takes  the  greatest 
pleasure  in  those  forms  of  intellectual 
activity, — within  the  field  in  -which  his 
natural  talents  and  education  fit  him  to 
work,  of  course, — that  present  the 
greatest  difficulties.  But  the  numbers 
of  those  to  whom  the  results  of  these 
activities  are  intelligible  are,  in  general, 
inversely  proportional  to  the  difficulties 
of  obtaining  them.  Thus  it  comes 
about  that  many  of  the  old  philoso- 


phers  are  best  known  by  those  of  their 
works  in  which  they  themselves  did  not 
take  the  greatest  interest.  Thales,  the 
first  of  the  Greek  philosophers,  the  first 
of  the  "seven  wise  men"  of  Greece, 
was  also  the  first  Greek  mathematician. 
Aristotle  was  a  physicist,  but  he  was 
also  the  first  to  enunciate  the  principle 
of  continuity  by  the  introduction  of  the 
idea  of  an  "infinitesimal,"  which  idea 
was  developed  by  Cavalieri,  Kepler, 
and  others,  and  led,  finally,  in  the 
hands  of  Leibnitz  and  Newton,  to  the 
invention  of  the  infinitesimal  calculus. 
Plato  was  a  zealous  promoter  of  mathe- 
matics among  the  Greeks.  Archimedes, 
although  a  physicist,  was  called  by  his 
immediate  successors  the  "great  mathe- 
matician." Kepler  was  a  mathema- 
tician as  well  as  an  astronomer.  Fin- 
ally, Descartes,  Leibnitz  and  Newton 
were  pre-eminently  mathematicians;  in 
fact,  from  a  certain  point  of  view,  I 
should  b«  inclined  to  consider  Des~ 


cartes  the  greatest  mathematician  that 
ever  lived. 

I  have  said  nothing  of  those  who  are 
known  only  as  mathematicians  and,  I 
may  almost  say,  are  known  only  to 
mathematicians.  My  object  has  been 
to  lay  the  foundation  for  my  opinion 
that  Omar  was  probably  above  all  a 
pure  mathematician.  The  distinction 
that  is  commonly  made  between  pure 
and  applied  mathematics  is  somewhat 
inconsistent.  Applied  mathematics  is 
not  a  branch  of  learning.  It  is  mathe- 
matics as  applied  to  practical  purposes. 
The  only  conceivable  reason  for  dis- 
tinguishing it  from  so-called  pure 
mathematics  is  that  the  concepts  to 
which  the  application  is  made  are  more 
or  less  necessarily  associated  with  other 
concepts  to  which  mathematics  is  not 
applicable.  There  is  but  one  mathe- 
matics, namely,  pure  mathematics, 
which,  however,  ha*  many  forms. 
Most  forms  or  branches  of  mathemat- 


fi 


ics  have  practical  applications.  Gauss, 
called  by  his  contemp cries  "princeps 
mathematicorum,"  himself  an  astron- 
omer by  profession,  praised  the  theory 
of  numbers  as  having  one  great  advan- 
tage over  all  other  branches  of  mathe- 
matics in  that  it  had  no  conceivable 
application  to  practical  purposes. 

The  only  mathematical  work  of 
Omar  with  which  we  are  acquainted  is 
his  "algebra."  Algebra  is  the  "soul" 
of  modern  mathematics;  in  its  original 
form  it  is  that  branch  of  mathematics 
that  deals  with  unknown  numbers. 
The  name  algebra  is  derived  from  "al 
gibr  w'al  mukhabala"  the  title  of  every 
Saracen  work  on  the  subject  since  the 
time  of  Abu  Jafar  Muhammed  ibn 
Musa  al  Khwarizmi  (circa  A.D.  825), 
who  was  long  supposed  to  have  in- 
vented the  subject.  But  we  now  know 
that  Al  Khwarizmi' s  work  is  simply  a 
translation  of  the  "dptOixeTtx-rj"  of  Dio- 
phantos  of  Alexandria  (circa  A.D. 


275).  Omar  was  one  of  a  long  series 
of  Saracen  algebraists  who  followed 
more  or  less  closely  in  the  track  of  Dio- 
phantos  and  Al  Kwarizmi.  Woepcke, 
in  his  French  translation  of  Omar's  al- 
gebra, finds  in  it  traces  of  the  influence 
of  Diophantos,  but,  he  says,  "these  are 
found  also  in  Muhammed  ibn  Mousa, 
and  there  exists  no  historical  datum 
that  proves  that  at  the  time  of  this 
algebraist  Diophantos  was  known  to 
the  Arabs."  But  we  know  better  now. 
In  algebra  as  the  science  of  unknown 
numbers,  it  is  necessary  to  have  some 
method  of  designating  the  unknown  in 
any  particular  question,  as  well  as  its 
positive  and  negative  powers.  Dio- 
phantos used  symbols  to  represent 
these,  but  his  symbols  are  simply  ab- 
breviations of  the  names  by  which  he 
called  the  corresponding  numbers  and 
in  the  text  stand  for  these  names 
rather  than  for  the  numbers.  The 
Saracen  mathematicians,  including 


8 


Omar,  adopted  translations  of  Dio 
phantos's  names  and  got  along  without 
symbols.  Thus  Omar  gives  a  certain 
equation  as  "a  cube  and  squares  are 
equal  to  roots  and  a  number,"  i.e.  x8  -f- 
ax2  =  bx  -f-  c.  He  calls  the  successive 
positive  powers  of  the  unknown  "root" 
or  "side,"  "square,"  "cube,"  "square- 
square,"  "square-cube,"  "cube-cube," 
etc.  and  the  successive  negative  powers 
(reciprocals  of  the  positive  powers) 
"part  of  root,"  "part  of  square,"  etc., 
as  Diophantos  did.  But  all  Omar's 
demonstrations  are  given  in  geomet- 
rical form,  which  was  the  standard 
form  among  the  Greeks;  in  fact,  the 
very  names  we  have  mentioned  are 
borrowed  from  geometry.  Moreover, 
Omar  solves  his  equations  by  means 
of  the  intersections  of  conic  sections; 
that  is,  he  solves  a  typical  form  of  the 
equation  under  consideration  in  this 
way  and  then  modifies  the  solution  to 
suit  the  particular  equation.  He  is  very 


systematic  throughout,  prefacing  each 
section  by  such  lemmas  as  he  will  have 
to  use. 

Omar's  greatest  original  contribution 
to  algebra  is  the  complete  classification 
of  the  cubic  equation,  a  classification 
that  he  recognizes  as  applicable  to 
equations  of  every  degree.  He  be- 
lieved that  cubic  equations  could  not 
be  solved  by  calculation,  but  that  one 
must  be  satisfied  with  the  construction 
of  solutions  by  intersecting  conies.  In 
the  discussion  of  the  several  classes  he 
sometimes  overlooks  particular  cases. 
Thus,  he  fails  to  see  that  an  equation 
of  the  form  x8  -j-  bx  =  ax2  -j-  c  may 
have  three  positive  real  roots.  Again, 
he  lost  many  roots  by  using  only  one 
branch  of  an  hyperbola  in  his  con- 
struction. And  he  was  not  very  exact 
in  the  investigation  of  the  numerical 
values  that  the  several  coefficients  must 
have  in  order  that  the  equation  of  one 
or  other  type  should  give  real  inter- 


10 


sections  of  the  conies.  He  considered 
biquadratic  equations  to  be  unsolvable 
by  geometric  constructions. 

But  these  faults  are  of  little  conse- 
quence in  comparison  with  the  re- 
markably great  advance  Omar  made 
in  algebra  by  treating  equations  of 
degree  higher  than  the  second,  and  by 
having  classified  them.  He  was  the 
only  mathematician  of  any  nation  be- 
fore 1,100  who  distinguished  trinomial 
cubic  equations  from  tetranomial, 
forming  two  groups  of  the  former  ac- 
cording as  the  term  of  the  2nd  or  1st 
degree  was  wanting,  and  two  groups 
of  the  latter  according  as  the  sum  of 
3  terms  was  equal  to  one  term  or  the 
sum  of  2  terms  equal  to  the  sum  of 
two  others. 

Apparently,  also,  he  considered  the 
binomial  theorem  for  positive  integral 
exponents.  He  says:  "I  have  taught 
how  to  find  the  sides  of  the  square- 
square,  of  the  square-cube,  of  the  cube- 


11 


cube,  etc.  to  any  extent,  which  no  one 
had  previously  done."  This  theorem 
he  used,  apparently,  for  the  purpose  of 
extracting  roots  after  the  manner  of 
the  Hindus.  Omar  incidentally  solved 
the  geometrical  problem:  to  construct 
an  equilateral  trapezoid  whose  base 
and  sides  are  of  the  same  given  length 
and  whose  area  is  given, — a  problem 
that  he  reduced  to  the  solution  of  the 
equation  x4  -f-  bx  =  ax8  -j-  c. 

In  the  year  1 079  Omar  corrected  the 
calendar.  He  grouped  the  years  in 
cycles  of  33  years  each,  giving  each 
common  year  365  days  and  making 
every  fourth  year  a  leap-year  of  366 
days  throughout  each  cycle;  that  is. 
each  cycle  of  33  years  contained  8 
leap-years  and  there  was  an  interval  of 
5  years  from  the  beginning  of  the  last 
leap-year  of  any  cycle  to  the  beginning 
of  the  first  leap-year  of  the  next  cycle. 
This  makes  the  average  length  of 
Omar's  solar  year  365«  5h  49»  5«.45, 


12 


which  is  less  by  6.55  seconds  than  the 
average  length  of  the  Gregorian  year. 
According  to  the  best  modern  calcula- 
tions, the  Gregorian  average  year  is  too 
long  by  25.557  seconds  and  Omar's 
average  year  is  too  long  by  only 
19.007  seconds.  That  is,  one  leap  year 
ought  to  be  omitted  from  Omar's  cal- 
endar every  4545  years,  whereas  the 
Gregorian  calendar  ought  to  omit  one 
leap-year  every  3381  years.  This 
means  that  Omar's  calendar  was  one- 
third  more  accurate  than  the  calendar 
we  use  today.  However,  all  people 
that  use  the  solar  year  would  probably 
find  it  more  convenient  to  omit  three 
leap-years  out  of  400  years  than  to 
group  the  years  in  cycles  of  33. 

All  things  considered,  I  am  inclined 
to  think  that  Omar  Khayyam  was  the 
most  original  and,  therefore,  the  great- 
est of  the  Saracen  mathematicians. 


13 


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